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Unformatted text preview: M99/510!H(2) INTERNATIONAL BACCALAUREATE
BACCALAUREAT INTERNATIONAL
BACHILLERATO INTERNACIONAL
MATHEMATICS Higher Level
Wednesday 5 May 1999 (morning) Paper 2 2 hours 30 minutes This examination paper consists of 2 sections, Section A and Section B.
Section A consists of 4 questions. Section B consists of 4 questions. The maximum mark for Section A is 80. The maximum mark for each question in Section B is 40. The maximum mark for this paper is 120. INSTRUCTIONS TO CANDIDATES Do NOT open this examination paper until instructed to do so. Answer all FOUR questions from Section A and ONE question from Section B. Unless otherwise stated in the question, all numerical answers must be given exactly or to three
signiﬁcant ﬁgures as appropriate. EXAMINATION MATERIALS Required: IB Statistical Tables
Millimetre square graph paper
Calculator Ruler and compasses Allowed:
A simple translating dictionary for candidates not working in their own language. 229-282 12 pages Trigonometrical identities: Integration by parts: Standard integrals: Statistics: If (xl,x2,.. FORMULAE sin(a + [3): sin 0: cos {3 + cosa sin ,8 cos(a + ﬂ) = cos 0: cos ﬂ — sin a sin [3 tan(a+l3)= tana+tanﬁ
l—tanatanﬁ
sina+sinﬁ=23ina+ﬁcosa—ﬁ
2 2
sina—sinﬁ=2cosa+ﬁsina_ﬁ
2 2
cosa+cosﬁ=2cosa+ﬁcosa_ﬁ
2 2
cosa—cosﬁ=25ina:ﬁsin3;a €0829=200329—1=1—28in29 =c0526—sin26 _ 2
Iftan§=tthensin6= 2’2 andcosfii=1 [2
2 l+t 1+:
u—d—vdx=uv-— vividx
dx dx dx 1 x
2 2 =—arctan—+c
x +a a a dx . x
= arcsm——+c (|x|< a)
:jaz —x2 ‘1 .,xn) occur with frequencies (f1 ,f2 ,. standard deviation 3 are given by m Binomial distribution: 229-282 _ Efﬁ- : s = zfxxi — m)2 2f; ’ if} ’ px = [n] p"(l-p)"", x =0,1, 2,...,n
x 1,2,... M9915 10/H(2) .., 1;) then the mean m and — 3 — M99/510/H(2) A correct answer with no indication of the method used will normally receive no marks. You are
therefore advised to show your working. SECTION A Answer all FOUR questions from this section. 1. [Maximum mark: 22] The coordinates of the points P, Q, R and S are (4,1 ,—l), (3,3,5),
(1 ,0 , 2c) , and (I , l ,2) , respectively. —> —->
(a) Find the value of c so that the vectors QR and PR are orthogonal. [7 marks ] For the remainder of the question, use the value of c found in part (a) for
the coordinate of the point R . -—> -—>
(b) Evaluate PS x PR. [4 marks] (c) Find an equation of the line l which passes through the point Q and is a
parallel to the vector PR. [3 marks]
((1) Find an equation of the plane it which contains the line l and passes
through the point S. [4 marks]
(c) Find the shortest distance between the point P and the plane it. [4 marks] 2. [Maximum mark: 20] (i) The ratio of the ﬁfth term to the twelfth term of a sequence in an arithmetic progression is If each term of this sequence is positive, and the product of the ﬁrst term and the third term is 32, ﬁnd the sum of the ﬁrst 100 terms
of this sequence. [7 marks] (ii) Let x and y be real numbers, and a) be one of the complex solutions of the
equation 23 = 1 . Evaluate: (a) l + a) + (02 [2 marks] (b) (1'01 + ‘02)“) (00235 + my) [4 marks] (iii) Using mathematical induction, prove that the number 22”— 3n n l is
divisibleby9,forn=1,2,.... . [7marks] 229-282 Turn over 3. [Maximum mark: 13] (i) A new blood test has been shown to be effective in the early detection of a disease. The probability that the blood test correctly identiﬁes someone
with this disease is 0.99, and the probability that the blood test correctly
identiﬁes someone without that disease is 0.95. The incidence of this
disease in the general population is 0.0001 . A doctor administered the blood test to a patient and the test result
indicated that this patient had the disease. What is the probability that the
patient has the disease? (ii) The quality control department of a company making computer chips knows that 2% of the chips are defective. Use the normal approximation to
the binomial probability distribution, with a continuity correction, to ﬁnd
the probability that, in a batch containing 1000 chips, between 20 and 30
chips (inclusive) are defective. 4. [ Maximum mark: 25] 229-282 (i) Consider the functionfzxr—nc— xzfor—I Exék, where l <k $3. (a) Sketch the graph of the function f. (b) Find the total ﬁnite area enclosed by the graph off, the x~axis and
the line x =k. (ii) Give exact answers in this part of the question. The temperature g(t) at time t of a given point of a heated iron rod is
given by lnt 3(‘)=$, where t > 0. _ (a) Find the interval where g’(t) > 0 . (b) Find the
go) <0. where interval the interval where g”(t)>0 and (c) Find the value of t where the graph of g(t) has a point of inﬂexion.
(d) Let t* be a value of t for which g’(t*) = 0 and g"(t*) < 0. Find r* . (e) Find the point where the normal to the graph of g(t) at the point
(1* , g(t*)) meets the t-axis. M99/510/H(2) [6 marks] [7 marks ] [3 marks] [4 marks] [4 marks] [5 marks]
[3 marks] [3 marks] [3 marks ] - 5 — M99!510/H(2) SECTION B Answer ONE question from this section. Abstract Algebra
5. [Maximum mark: 40] (i) Let S be the group of permutations of {1 , 2 , 3} under the composition of
permutations. ' (a) What is the order of the group S? [2 marks] (b) Let p0 , p1 , p2, be three elements of S, as follows: _123_123p_123
p°123’p‘231’2312' List the other elements of S and show that S is not an Abelian group. [4 marks] (0) Find a subgroup of S of order 3. [2 marks ]
(ii) (a) Let A be the set of all 2 X 2 matrices of the form [ :7 b], where
"- a
a and b are real numbers, and a2 + I)2 ah 0. Prove that A is a group
under matrix multiplication. [10 marks]
(b) Show that the set2M= 1 0,1 0"1 0,“ 0
0 1 0 —1 0 l 0 ——1
forms a group under matrix multiplication. [5 marks]
(0) Can M have a subgroup of order 3 ? Justify your answer. I [2 marks]
(iii) (a) Deﬁne an isomorphism between two groups (G, o) and (H, o) . [2 marks] (b) Let c and e’ be the identity elements of groups G and H respectively.
Let f be an isomorphism between these two groups. Prove that
f(e)=e'- [4 marks] (c) Preve that an isomorphism maps a ﬁnite cyclic group onto another
ﬁnite cyclic group. [4 marks] (d) Consider Z4, the additive group of integers modulo 4. Prove that 224 is cyclic. Is 224 isomorphic to M in part (ii)(b) under matrix
multiplication? Justify your answer. [5 marks] 229-282 Turn over — 6 — M99/510/H(2) Graphs and Trees 6. [Maximum mark: 40]
(i) (a) Give a deﬁnition of the adjacency matrix of a directed graph G. [2 marks] (b) What is represented by the sum of the entries in the ith row of the
adjacency matrix of G ? [2 marks] (c) What is represented by the sum of the entries of the jth 'column of the
adjacency matrix of G ? [2 marks] (ii) (a) Draw a directed graph whose adjacency matrix A is given by the following:
1 O 0 1 0
l 0 0 l 1
A = l l 0 0 0
l 0 0 0 0
0 1 0 0 0 [3 marks]
(b) How many paths of length 2 are there from vertex 122 to vertex v4 [2 marks]
(iii) Let It" be the complete graph of order n and KM be a bipartite graph of
orders m and n .
(3) Explain the following, giving one example of each:
(i) K5 , the complete graph of order 5 , [3 marks]
(ii) a bipartite graph 1c” . [2 marks]
(b) Show that K33 has a Hamiltonian cycle, giving appropriate reasons. [3 marks] (This question continues on the following page) 229-282 — 7 — M99/510/H(2) (Question 6 continued)
(iv) The following ﬂoor plan shows the ground level of a new home. Is it
possible to enter the house through the front door and exit through the rear door, going through each internal doorway exactly once? Give a reason for
your answer. [7 marks ] Front door E3 Rear door (v) (a) Prove that if two graphs are isomorphic, they have the same degree
sequence. [3 marks] (b) Are the following graphs isomorphic? Justify your answer. [3 marks] (This question continues on the following page) 229-282 Turn over — 8 — M9915 10/H(2) (Question 6 continued) (vi) Apply Prim’s algorithm to the weighted graph given below to obtain the
minimal spanning tree starting with the vertex A . Find the weight of the minimal spanning tree. [8 marks] 229-282 _ 9 _ M99/510/H(2)
Statistics 7. [Maximum mark: 40] (i) A supplier of c0pper wire looks for ﬂaws before despatching it to
customers. It is knowu that the number of ﬂaws follow a Poisson
probability distribution with a mean of 2.3 ﬂaws per metre. (a) Determine the probability that there are exactly 2 ﬂaws in 1 metre of
the wire. [3 marks] (b) Determine the probability that there is at least one ﬂaw in 2 metres of
the wire. [3 marks] (ii) A market research company has been asked to ﬁnd an estimate of the mean
hourly wage rate for a group of skilled workers. It is known that the
population standard deviation of the hourly wage of workers is $4.00 .
Using a conﬁdence interval for the mean, determine how large a sample is
required to yield a probability of 95% that the estimate of the mean hourly
wage is within $0.25 of the actual mean. [10 marks] (iii) The administration of a certain university wants to examine whether there
is a signiﬁcant difference between the wages of humanities and science
students employed within the university for temporary work. Random
samples of wages are tabulated below: Weekly mean wage Standard deviation Test, at the 2% level of signiﬁcance, whether the sample shows a difference
between the wages. I [12 marks] (Tins question continues on the following page) 229-282 - Tum over -10.. (Question 7 continued) 229-282 (iv) A car manufacturer wants to know if there is a relationship between the M99f510/H(2) cost of a new vehicle and the average number of complaints. The company ' checks its records of complaints and collects the following data from a
random sample of 1000 vehicles. Number of complaints
mm 100 90 10
150 260 50
50 250 40 a $30 001
$15 001—$30 000
S $15 000 At the 5% level of signiﬁcance, is there a relationship between the cost of a
new vehicle and the number of complaints? [12 marks] -1]... Analysis and Approximation 8. [Maximum mark: 40] (i) (a) (13) (ii) (a) (C) (iii) (a)
03) 229-282 Show that there is zero of the function f(x)=x3—3x—5 in the
interval 1% x S 3 . With the initial estimate x0: 2 , use the Newton-Raphson method to
ﬁnd the solution of the equation f(x)=0 in the interval 1 S xi 3 ,
accurate to 10‘5 . Using the trapezium rule and Simpson’s rule with 6 sub-intervals, L. where g(x) is given at seven points by the following table. evaluate the integral g(x) dx, 0.97260 0.89483 0.77880 0.64118 0.49935 0.36789
(13) Find the error estimate for Simpson’s rule in terms of g‘4)(x). When | g“)(x) | s 6 , determine the number of subintervals required to
use Simpson’s rule to obtain a value for the above integral, which is
correct to ﬁve decimal places. State the mean value theorem. A given function h satisﬁes all the requirements of the mean value
theorem for 0 S. x S. 7. If h(0)=—4 and lh' (x) | s 10 use the mean value theorem to Show that h(x) 2 —74 , for 0 S x s. 7 . (This question continues on the following page) M99!510/H(2) [2 marks] [6 marks] [6 marks]
[2 marks] [6 marks] [2 marks] [4 marks] Turn over — 12 — M99!510lH(2) (Question 8 continued) (iv) Test the convergence or divergence of the following inﬁnite series,
indicating the tests used to arrive at your conclusion: (a) E k3: l { 3 marks]
k=l
°° 1 b E 4 k ( ) k=2 kﬂn 103 [ mar s]
°° k (c) (—1)“I [5 marks]
2:4 k2 +1 229-232 ...
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